BY I T . \-, lRIETCAILF
In a recent paper’ I tried to make it clear that \-an der n’aals’ equation is an expression of an equilibrium between a RT opposing pressures. It may be written Pi .? = -
+
-L9
h
(Equation A) or PIi- P? = Pi. PIis the external pressure applied to the substance
P,( = :2)
is the cohesive pressure
due to the attraction betm een the molecules. These two pressures act together in tending to decrease the x-olume.
(=
P3
y,)is a pressure nhich tends to increase the volume
of the mass It is due, according to the kinetic theory, to the total effect per unit area of the blows of the indiyidual molecules. I have called it the elastic pressure. Xs shown in the previous paper. this term has the dimensions of a pressure, since from the ideal gas equation RT has the dimensions of p : ~and ;’ - E expresses a \ o h m e . It Seems therefore to be a necessary deduction from the kinetic theory and from the nature of the terms themseh-es that \-an der 17-aals’ equation is an expression of an equilibrium between opposing pressures, which determines the volume of the mass. By the term “volume coefficient” of a pressure in this I dP paper I shall mean p ’ d., , that is, the rate of change of the
zj
pressure with T-olume relative to its own value. By the “rate of change with volume” of a pressure I mean
that
is, its absolute rate of change. I n the former paper the terms “density coefficient” and “rate of change with density” were used instead of volume coefficient and rate of change with volume. The relation between the two is evident, since c is the reciprocal of D if unit mass is taken. In the present Jour. Phys Chem., 19,705 (1915).
paper it is found more convenient to express the relations in terms of \-nlunie instead of pressure. The principal conclusions of the previous paper, expressed in terms of s-olume instead of pressure, were as follows : I . The s-olume of a gas or liquid is determined by an equilibrium of pressures, as ab0L-e. 2 . In a stable liquid the ;oliriizi, coL-@ciciif of the pressure P3 is greater than that of the pressure P.'.As the liquid increases in s-olume along a volume-temperature curve however. the ratio of the volume coefficient of P2 to that of P3 increases; i t becomes equal to unity at a certain definite volume; and thereafter the coefficient of P:! is increasingly greater than that of PB. 3 . In a stable liquid the rate o f slz~71igc ;,it/t Lolzme of PI P1 is less than that of P?. As the liquid increases in volume along the .volume-temperature curs-e however, the ratio of the first rate to the second increases; it becomes equal to unity at c, the maximum point of the curve; beyond c it continues to increase to a maximum a t d , after which it decreases, passing through the value unity again at the minimum point c, and beyond c it is a decreasing proper fraction. 4. The points c and c therefore are points at which the rate of change with volume of PI P2 is equal to that of Pa;at which therefore the coefficient of expansion becomes infinite ; and a t which therefore the nature of the equilibrium changes from stable to unstable and :'ice ; c i s u . On curves plotted at successiT7ely higher constant external pressures, these tn-o points must, as a necessary result of pressure equilibrium concept, approach nearer and nearer to each other until they finally coincide at the critical temperature. In the previous paper I tried t o show in a purely qualitative way, and largely from phj-sical considerations, that the above propositions are necessary deductions from the kinetic theorj-. as applied t o van der TT-aals' equation and to the phenomena in question, and that they offer a simple mechanical explanation of all the phenomena. I called attention also
+
+
to the fact that this pressure equilibrium concept gives a clear insight into the mechanical cause of various phenomena that are not otherwise easily explained. As examples, the explanation of osmotic flow, and of the fact that the vapor tension of a liquid is decreased by dissolving in it a non-volatile substance, were cited. It is not difficult t o explain, in a concrete qualitative way from physical considerations, the mechanical cause of the curious relations described above. In doing this we would ha\-e to keep in mind that the cohesive pressure is only a part of the pressure on the left of the equation, while the elastic pressure is the entire pressure on the right; also the fact that the cohesive pressure
(
f2)
varies in-
versely as the square of the whole volume, while the elastic pressure
(,”’,>
varies inversely as tiic first power of the
volume J - b, and directly as the absolute temperature, and the physical reasons for this. This is connected with the fact that the cohesix-e force between the molecules is a function of the average distance between the centers of adjacent molecules, while the elastic force, depending a t any given temperature on the mean free path, varies as some function of the average distance between the surjaces of adjacent molecules. TTe should bear in mind also that in liquids the average distance between the surfaces of adjacent molecules ion van der n’aals’ assumption that the molecules are perfectly elastic, incompressible spheres) is ordinarily less than one-half the diameter of the molecule. (See Boynton’s “Kinetic Theory,” Chapter 8.) W’e will not elaborate this point a t present. It is possible to prove the above propositions to be rigid mathematical deductions from \-an der Uraals’ equation, and it is the purpose of the present paper to do this. As in the former paper, the discussion will be practically confined to the series of volume-temperature curves plotted a t successively higher constant external pressures. (See Fig. I . ) An entirely
similar analysis applies to the volume-pressure curves plotted at constant temperatures.
V Fig.
I
From Equation A, The rate of change with voluine of Pz = (I)
2a ~
+ Pz= za +$) = -2.
The rate of change with volume of PI d dv --- (PI
(P1 constant.)
The rate of change w i t h volume of Ps = I dP2 The volume coefficient of Pz = . -_
Pz dv
The volume coefficient of PI
+ P2 =
The volume coefficient of P3 = I ap3 RT p B 'av
E-b
=
2a v2 -.-
v3 a
- -- I
-
= -2
8'
RT v-bb' It is to be noted that the rates of change with volume of PI P2 and of P3 are the same, since P is constant, but their volume coeficients are different. ( 2 ) Equating the expressions in Paragraph I for the
+
(v-b)'
'
V a n der lit'aals' Equatioiz
181
2 I - - Vvolume coeficients of PPrand P3,- ;- b, or
D =
2b.
Theref ore When v < 2b, the volume coefficient of Pz< the volume coefficient of Pa (numerically). When v = 2b, the volume coefficient of Pz= the volume coefficient of P3 When v > 2b, the volume coefficient of P2> the volume coefficient of Ps (numercially), and as i' increases this excess increases. Stated differently, When E < z b , the ratio of the volume coefficient of PZto that of
P3 =
= 2b, the Z(U - b ) = I
ratio of the volume coefficient of Pz to that of
When v
(y-)
When v > 2b, the ratio of the volume coefficient of Pz t o that of z(iir-b) PB= (7 > I , and -) 2 ( ~ b) as v increases the ratio increases, approaching the
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